Maximal and stabilizing Hermitian solutions for discrete-time coupled algebraic Riccati equations

Discrete-time coupled algebraic Riccati equations that arise in quadratic o ptimal control and H-infinity-control of Markovian jump linear systems are considered. First, the equations that arise from the quadratic optimal cont rol problem are studied. The matrix cost is only assumed to be hermitian. C onditions for the existence of the maximal hermitian solution are derived i n terms of the concept of mean square stabilizability and a convex set not being empty. A connection with convex optimization is established leading t o a numerical algorithm. A necessary and sufficient condition for the exist ence of a stabilizing solution (in the mean square sense) is derived. Suffi cient conditions in terms of the usual observability and detectability test s for linear systems are also obtained. Finally, the coupled algebraic Ricc ati equations that arise from the H-infinity-control of discrete-time Marko vian jump linear systems are analyzed. An algorithm for deriving a stabiliz ing solution, if it exists, is obtained. These results generalize and unify several previous ones presented in the literature of discrete-time coupled Riccati equations of Markovian jump linear systems.